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What does non-dimensionalization tell us about the spreading of Myxococcus xanthus ?

What does non-dimensionalization tell us about the spreading of Myxococcus xanthus ?. Angela Gallegos University of California at Davis, Occidental College Park City Mathematics Institute 5 July 2005. Acknowledgements. Alex Mogilner, UC Davis

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What does non-dimensionalization tell us about the spreading of Myxococcus xanthus ?

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  1. What does non-dimensionalization tell us about the spreading of Myxococcus xanthus? Angela Gallegos University of California at Davis, Occidental College Park City Mathematics Institute 5 July 2005

  2. Acknowledgements • Alex Mogilner, UC Davis • Bori Mazzag, University of Utah/Humboldt State University • RTG-NSF-DBI-9602226, NSF VIGRE grants, UCD Chancellors Fellowship, NSF Award DMS-0073828.

  3. OUTLINE • What is Myxococcus xanthus? • Problem Motivation: • Experimental • Theoretical • Our Model • How non-dimensionalization helps!

  4. OUTLINE • What is Myxococcus xanthus? • Problem Motivation: • Experimental • Theoretical • Our Model • How non-dimensionalization helps!

  5. Rod-shaped bacteria Myxobacteria are:

  6. Myxobacteria are: • Rod-shaped bacteria • Bacterial omnivores: sugar-eaters and predators

  7. Myxobacteria are: • Rod-shaped bacteria • Bacterial omnivores: sugar-eaters and predators • Found in animal dung and organic-rich soils

  8. Why Myxobacteria?

  9. Why Myxobacteria? • Motility Characteristics • Adventurous Motility • The ability to move individually • Social Motility • The ability to move in pairs and/or groups

  10. Why Myxobacteria? Rate of Spread Non-motile 4 Types of Motility Adventurous Mutants Social Mutants Wild Type

  11. OUTLINE • What is Myxococcus xanthus? • Problem Motivation: • Experimental • Theoretical • Our Model • How non-dimensionalization helps!

  12. Experimental Motivation • Experimental design • Rate of spread r0 r1

  13. Experimental Motivation *no dependence on initial cell density *TIME SCALE: 50 – 250 HOURS (2-10 days) Burchard, 1974

  14. Experimental Motivation * TIME SCALE: 50 – 250 MINUTES (1-4 hours) Kaiser and Crosby, 1983

  15. Experimental Motivation

  16. OUTLINE • What is Myxococcus xanthus? • Problem Motivation: • Experimental • Theoretical • Our Model • How non-dimensionalization helps!

  17. Theoretical Motivation • Non-motile cell assumption • Linear rate of increase in colony growth • Rate dependent upon both nutrient concentration and cell motility, but not initial cell density r Gray and Kirwan, 1974

  18. Problem Motivation

  19. Problem Motivation

  20. Problem Motivation • Can we explain the rate of spread data with more relevant assumptions?

  21. OUTLINE • What is Myxococcus xanthus? • Problem Motivation: • Experimental • Theoretical • Our Model • How non-dimensionalization helps!

  22. Our Model • Assumptions • The Equations

  23. Our Model • Assumptions • The Equations

  24. Assumptions • The cell colony behaves as a continuum

  25. Assumptions • The cell colony behaves as a continuum • Nutrient consumption affects cell behavior only through its effect on cell growth

  26. Assumptions • The cell colony behaves as a continuum • Nutrient consumption affects cell behavior only through its effect on cell growth • Growth and nutrient consumption rates are constant

  27. Assumptions • The cell colony behaves as a continuum • Nutrient consumption affects cell behavior only through its effect on cell growth • Growth and nutrient consumption rates are constant • Spreading is radially symmetric r1 r2 r3

  28. Assumptions • The cell colony behaves as a continuum • Nutrient consumption affects cell behavior only through its effect on cell growth • Growth and nutrient consumption rates are constant • Spreading is radially symmetric r1 r2 r3

  29. Our Model • Assumptions • The Equations

  30. The Equations • Reaction-diffusion equations • continuous • partial differential equations

  31. The Equations: Diffusion • the time rate of change of a substance in a volume is equal to the total flux of that substance into the volume J(x0,t) c J := flux expressionc := cell density J(x1,t)

  32. The Equations: Reaction-Diffusion • Now the time rate of change is due to the flux as well as a reaction term J(x0,t) c J := flux expressionc := cell density f := reaction terms J(x1,t) f(c,x,t)

  33. The Equations: Cell concentration • Flux form allows for density dependence: • Cells grow at a rate proportional to nutrient concentration

  34. The Equations: Cell Concentration c := cell concentration (cells/volume) t := time coordinate D(c) := effective cell “diffusion” coefficient r := radial (space) coordinate p := growth rate per unit of nutrient (pcn is the amount of new cells appearing) n := nutrient concentration (amount of nutrient/volume)

  35. The Equations: Cell ConcentrationThings to notice flux terms reaction terms: cell growth

  36. The Equations: Nutrient Concentration • Flux is not density dependent: • Nutrient is depleted at a rate proportional to the uptake per new cell

  37. The Equations: Nutrient Concentration n:= nutrient concentration (nutrient amount/volume) t := time coordinate Dn:= effective nutrient diffusion coefficient r := radial (space) coordinate g := nutrient uptake per new cell made (pcn is the number of new cells appearing) p := growth rate per unit of nutrient c := cell concentration (cells/volume)

  38. The Equations: Nutrient Concentration Things to notice: flux terms reaction terms: nutrient depletion

  39. The Equations: Reaction-Diffusion System

  40. Our Model: What will it give us?

  41. OUTLINE • What is Myxococcus xanthus? • Problem Motivation: • Experimental • Theoretical • Our Model • How non-dimensionalization helps!

  42. Non-dimensionalization: Why?

  43. Non-dimensionalization: Why? • Reduces the number of parameters • Can indicate which combination of parameters is important • Allows for more computational ease • Explains experimental phenomena

  44. Non-dimensionalization:Rewrite the variables where are dimensionless, and are the scalings (with dimension or units)

  45. What are the scalings? is the constant initial nutrient concentration with units of mass/volume.

  46. What are the scalings? is the cell density scale since g nutrient is consumed per new cell; the units are:

  47. What are the scalings? is the time scale with units of

  48. What are the scalings? is the spatial scale with units of

  49. Non-dimensionalization:Dimensionless Equations

  50. Non-dimensionalization: Dimensionless EquationsThings to notice: • Fewer parameters: p is gone, g is gone • remains, suggesting the ratio of cell diffusion to nutrient diffusion matters

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